Sums of random polynomials with differing degrees

Abstract

Let μ \mu and ν \nu be probability measures in the complex plane, and let p p and q q be independent random polynomials of degree n n , whose roots are chosen independently from μ \mu and ν \nu , respectively. Under assumptions on the measures μ \mu and ν \nu , the limiting distribution for the zeros of the sum p + q p+q was computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021), p. 124719] as n → ∞ n \to \infty . In this paper, we generalize and extend this result to the case where p p and q q have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of μ \mu and ν \nu , scaled by the limiting ratio of the degrees of p p and q q . Additionally, our approach provides a complete description of the limiting distribution for the zeros of p + q p + q for any pair of measures μ \mu and ν \nu , with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.